Families of $2$-weights of some particular graphs

TL;DR

This paper provides criteria to determine when a given family of positive real numbers can be realized as shortest path weights in specific classes of graphs, such as snakes, caterpillars, polygons, bipartite, complete, and planar graphs.

Contribution

It introduces a method to characterize families of distances that correspond to weighted graphs within particular graph classes.

Findings

Established criteria for families of distances in specific graph classes.

Applied the criteria to classes like snakes, caterpillars, and polygons.

Provided a framework for reconstructing weighted graphs from distance data.

Abstract

Let ${\cal G}=(G,w) $ be a positive-weighted graph, that is a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of positive real numbers; for any distinct vertices $i,j $, we define $D_{i,j}({\cal G})$ to be the weight of the path in $G$ joining $i$ and $j$ with minimum weight. In this paper we fix a particular class of graphs and we give a criterion to establish whether, given a family of positive real numbers $\{D_I\}_{I \in { \{1,...., n\} \choose 2}}$, there exists a positive-weighted graph ${\cal G} =(G,w) $ in the class we have fixed, with vertex set equal to $\{1,....,n\}$ and such that $D_I ({\cal G}) =D_I$ for any $I \in { \{1,...., n\} \choose 2}$. In particular, the classes of graphs we consider are the following: snakes, caterpillars, polygons, bipartite graphs, complete graphs, planar graphs.